Obviously, the units of the unsteady and convective term on the left do not correspont to the units of the diffusion term on the right. I think that rho has to be added to the diffusion term aswell, but I don't know how to do that.

Dividing the whole equation by rho is also not possible, because it is compressible.

really solves the dimension problem. BUT I'm not sure if this is allowed in respect to the underlying physics of the problem.

Can anybody say something about that?

In addition, the now calculated alphaN, which is initialized as 1 in the jet flow and 0 in the crossflow increases slightly above 1 (< 1.09) in regions of high density gradients and in other regions it also decreases slightly below 0 (> 0.9999). That should not be possible for how I wanted alphaN to be.

Does anybody know why that happens? The overall picture of the alphaN field looks allright. Could the mesh be the problem? It is still pretty coarse for testing reasons. Or should I devide alphaNEqn by density though?